Apparatus for analog to digital conversion

ABSTRACT

An input analog signal is converted to a digital output signal by an oversampled predictive DPCM coder which includes an n stage delay line in the feedback loop. The n delay line outputs are weighted by coefficients a i  . . . a n  selected according to the relationship ##EQU1## AND THEN SUMMED. Alternatively, the feedback loop may comprise a chain of n integrators arranged so that the signal fed back to the comparator is the sum of single, double, triple...n order integration. By so doing, the coder attenuates the signal power at the quantizer input while the attenuator coefficients are independent of the input signal statistics. 
     A similar technique may also be applied in an oversampled error feedback coder, which includes a feedback loop having an n stage delay line. Here again, the delay line outputs are weighted in accordance with the above relationship. Alternatively, a chain of n integrators may be used in the coder input, and an identical chain employed in the feedback loop. By so doing, the coder attenuates the coding noise power in the signal band while the coder design is rendered independent of the quantizing error statistics.

This application is a division of application Ser. No. 608,524 filedAug. 29, 1975.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to conversion of analog signalsto digital signals and, more particularly to both predictive and errorfeedback coders that sample the analog signal at generally high ratesand provide a differential pulse code modulation (DPCM) output.

2. Description of the Prior Art

Sophisticated digital integrated circuit technologies have made digitalsignal processors which perform a variety of functions a practicalalternative to conventional analog systems. The trend toward digitalprocessing is accompanied by many advantages, including preciselypredictable performance, the ability to share hardware over severalseparate channels without interchannel distortion, and the flexibilityof programming one processor to perform a variety of functions.

Since digital processors must typically interface with continuoussignals, analog/digital and digital/analog converters thus becomeincreasingly important system components. Many such converters have beendeveloped, including predictive coders which reduce the dynamic range ofthe input signal applied to the quantizer within the coder, or errorfeedback coders which shape the spectral distribution of the quantizingerror so as to reduce in-band noise. Both types of coders may make useof tapped delay lines in the feedback path, with the n outputs of thedelay line being fed through n separate attenuators having coefficientsa₁ . . . a_(n), and then summed. With this arrangement, varioustechniques are available for choosing appropriate attenuatorcoefficients. In each technique, it is desirable to minimize the numberof required quantizaton levels, so that the number of bits needed toadequately represent the signal being encoded is reduced. Certain of theprior art techniques for coefficient optimization have been developedprimarily for transmission systems. However, in such systems,oversampling, as desired in the present invention, cannot be used, sinceit results in an undesirably high transmission rate. Accordingly, onesubject of the present invention is to optimize the coefficientselection technique in cases where oversampling is acceptable.

Other prior art techniques known in the field of video transmission haveutilized moderate oversampling in conjunction with single or doubleintegration feedback networks. However, these systems have also beenlimited in the degree of quantizer simplification achieved. Accordingly,another object of the present invention is to retain the signalindependent feedback network as in video coders, while achieving a muchgreater reduction in the number of required quantization levels.

SUMMARY OF THE INVENTION

Each of the foergoing and additional objects are achieved in accordancewith the principles of the instant invention by providing, in bothpredictive and error feedback oversampled coders which include delaylines having n output stages, a series of n attenuators havingcoefficients selected according to the relationship ##EQU2##Alternatively, in the case of predictive coders, the feedback loop mayinclude a chain of n integrators arranged so that the signal fed back tothe comparator is the sum of single, double, triple . . . n orderintegration. In the case of error feedback coders, serial chains of nintegrators may be employed both in the coder input line, and in thefeedback loop. By virtue of the advantageous coder arrangement of thepresent invention, the number of levels required in the quantizer issignificantly reduced, while the feedback network is maintainedindependent of the input signal.

BRIEF DESCRIPTION OF THE DRAWING

The invention will be more clearly understood from a consideration ofthe following detailed description when read in light of theaccompanying drawing in which:

FIG. 1 is a block diagram of a prior art predictive coder;

FIG. 2 is a block diagram of one realization of the feedback filter ofFIG. 1;

FIG. 3 is a graph of both the mean square error signal and the reductionin the number of required quantizer levels in the coder of FIGS. 1 and2, when the coefficients in accordance with the invention are used;

FIG. 4 is a block diagram of a predictive coder constructed inaccordance with the principles of the present invention;

FIG. 5 is an alternative arrangement for the coder of FIG. 4;

FIG. 6 is a graph of the transfer function of the coders of FIGS. 4 and5 as a function of frequency;

FIG. 7 is a block diagram of a prior art error feedback coder;

FIG. 8 is a block diagram of an error feedback coder constructed inaccordance with the principles of the instant invention;

FIG. 9 is a block diagram of a third order integrator that may be usedin the coder of FIG. 8; and

FIG. 10 is a graph showing the reduction in the number of quantizationlevels achievable using the coder of FIG. 8.

DETAILED DESCRIPTION

As shown in FIG. 1, a prior art predictive coder typically includes aforward loop containing an analog to digital (A/D) converter such asquantizer 101 and a feedback loop which contains a feedback filter 102,to be discussed in more detail hereinafter. The output of feedbackfilter 102 is fed back to its input via line 103 and additive circuit104, which combines the output with the output of quantizer 101. Thefeedback filter 102 output is also supplied to subtractive circuit 105,thus completng the feedback circuit to the quantizer.

In operation, the quantizer 101 input signal e_(n) is the differencebetween the input signal s_(n) and the output f_(n) of filter 102.Quantizer 101, assumed to have equal quantization levels, has an outputE_(n) which approximates e_(n) by the nearest discrete quantizationlevel, i.e.,

    E.sub.n = e.sub.n + q.sub.n                                (1)

where q_(n) is the quantization error. From the above relationships, itis seen that the output of circuit 104 is given by:

    X.sub.n = E.sub.n + f.sub.n                                (2)

and the output of circuit 105 is given by:

    e.sub.n = s.sub.n - f.sub.n                                (3)

so that, from equations 1, 2 and 3:

    X.sub.n = (e.sub.n +f.sub.n) + q.sub.n = s.sub.n +q.sub.n. (4)

If the characteristis of feedback filter 102 are appropriately chosen,its output f_(n) is a good prediction of the next input signal samples_(n). As a result, e_(n) which represents the extrapolation error ofthe coder, is quantized in quantizer 101, instead of the input signals_(n), as would be the case in an A/D converter not using predictivefeedback. Specifically, if s_(p) and e_(p) represent the peak values ofs_(n) and e_(n), respectively, and it is assumed that s_(p) and e_(p)are proportional to their mean square values σ₅ ² and σ_(e) ², thenumber of quantization levels required in quantizer 101 is reduced by afactor

    θ = √σ.sup.2.sub.e / √σ.sup.2.sub.s (5)

relative to the number required in nonpredictive coders.

One known manner of arranging filter 102 to produce the above results isthe use of the delay line having a series of n stages such as stages201, 202, 203 and 204 as shown in FIG. 2. connected to the output ofeach stage 201 through 204 are n attenuators such as attenuators 211,212, 213 and 214 each having a coefficient a₁, a₂, a₃ . . . a_(n). Theoutputs of attenuators 211 through 214 are summed in additive circuits220, 221, 222 so that the Z-transform K(Z) of the entire filter is givenby: ##EQU3##

When the delay line arrangement of FIG. 2 is used in the coder of FIG.1, the mean square value of th quantizer input signal, neglectingquantizing errors, is given by: ##EQU4## wherein ψ_(m), theautocorrelation function of the input signal, is given by: ##EQU5## Thestandard prior art procedure for minimizng σ² _(e) is to simultaneouslysolve the N linear equations:

    ∂σ.sup.2.sub.e /∂ a.sub.i = 0 for i = 1.2 . . . N .                                                   (9)

while solution of equation (9) yields the filter coefficients a_(i), thecoefficients are functions of and depend upon th input samplecorrelation ψ_(e). However, such dependence is not desirable in manyapplications, since the input signals are nonstationary. Accordingly, analternative filter design approach is needed.

In accordance with the present invention, signal independent attenuatorcoefficients a_(i) are computed according to the relationship ##EQU6##This relationship is arrived at by first considering that the inputsample correlations ψ_(e) and the two-sided power spectral density G_(s)(f) of the input signal are related by: ##EQU7## where f_(s) is thesample rate and f_(o) is the input signal bandwidth. Using equation (11)in equation (7) σ² _(e) may be expanded as a power series in (f_(o)/f_(s))², since it is assumed that f_(s) >>2f_(o) due to oversampling.Such expansion yields the general form ##EQU8## where the β_(l) dependonly on the filter coefficients and N. By choosing the filtercoefficients in accordance with equation (10), β_(l) = 0 for l ≦ N-1 inequation (12), leaving the lowest order nonvanishing term in equation(12) as ##EQU9## Combining equations (13) and (5), it is seen that byusing the coefficients selected in accordance with equation (10), thenumber of quantization levels required is significantly reduced. Forexample, as shown in FIG. 3, for a three stage delay line (N=3) and fora sampling rate f_(s) of 20 times that of the input signal bandwidthf_(o), the number of quantization levels required in quantizer 101 isreduced by more than a factor of 64, as compared to the case of aquantizer without feedback. For the same ratio of f_(s) /f_(o), thereduction of levels for 4 and 5 stage delay lines are greater than 256and 512, respectively. Alternatively stated, it is seen from FIG. 3 thatby the advantageous choice of attenuator coefficients in accordance withthe present invention, σ² _(e) decreases by 6N dB per octave increase insampling rate, for an N stage delay line; quantization levels areaccordingly decreased by a factor of 2N for each doubling of thesampling rate.

Using the results of equation (10) in equation (6), it will be seenthat, in Z transform notation, the transfer characteristic of filter 102is given by:

    K(Z) =  1-(1-Z.sup.-1).sup.N.                              (14)

also, it will be seen that the transfer function T(Z) between the outputand input of filter 102, taking into account feedback line 103, is:##EQU10## Substituting equation (14) in (15), the output/input transferfunction is thus: ##EQU11## Since equation (16) may be written in theform ##EQU12## and since the transfer function of a single integrator isgiven by 1/(1-Z⁻¹), a coder using analog prediction rather than thepredictive filter of FIG. 2, may be constructed in accordance with theinvention as shown in FIG. 4. As before, the forward loop contains aquantizer 401 which is arranged to sample the analog input signalapplied thereto at a rate f_(s) much greater than the input signalbandwidth f_(o), and to provide at its output a digital signal E_(n),representative of said input signal. The digital signal is fed, in thefeedback loop, first to a digital to analog converter 402, and then to aserial chain of M analog integrators, such as integrators 410, 411, and412. The outputs of integrators 410, 411, and 412 are connected vialines 420, 421 and 422 to an adding circuit 430, such that the adderoutput consists of the sum of single, double, triple . . . M orderintegration. The feedback loop is completed by connecting the output ofadding circuit 430, which output is a good prediction of the inputsignal, to the negative input terminal of a comparator or differencingcircuit 440, the positive input to which is the analog input signal.

Inspection of the circuit of FIG. 4 reveals that the transfer functionbetween the output of converter 402 and the output of adder 430 isgiven, as desired, by equation (17). The initial Z⁻¹ term is built intothe time response of the integrators. The use of D/A converter 402 priorto the integrators 410 through 412 in the feedback loop of courserequires that the integrators be designed to operate on analog signals.This arrangement is, however, preferred over the use of digitalintegrators followed by a digital to analog conversion, since a veryhigh degree of converter precision would then be needed. However, whenthe coder is configured as shown in FIG. 4, converter 402 may beconstructed with a degree of precision only matching that of quantizer401, which as explained previously, is advantageously quite simple.

An alternative arrangement to the coder of FIG. 4 is shown in FIG. 5. Inthis embodiment, quantizer 501, converter 502, and differencing circuit540 are all equivalent to their FIG. 4 counterparts. Adder circuits 550and 551 are disposed after each integrator 510 and 511 in the serialchain of M integrators, with the exception only of the last integrator512. The output of converter 502 is simultaneously applied to the inputof the first integrator 510 in the chain, as well as to each addercircuit 550 and 551. As will be seen by inspection, the output of thelast integrator 512 is the sum of first, second, third . . . M orderintegration, as was the case for the coder of FIG. 4.

The advantages of the predictive coders of FIGS. 4 and 5 may be stillbetter appreciated by reference to the graph of FIG. 6, in which thecoder transfer function (the ratio of the coder output to input, in dBis plotted as a function of the radio f/f_(s) of the input signalfrequency to the sampling frequency. For a second order coder inaccordance with the invention, i.e., a coder having a feedback loop inwhich the output is the sum of single plus double integration, curve 601shows the coder transfer function, which is given by:

    [h(f)].sup.2 = 4(1-cos2πf/f.sub.s).sup.2 .              (18)

By way of comparison, curve 602 shows the transfer function of a priorart coder having two serially connected integrators in the feedbackloop. The transfer function in this case is given by: ##EQU13##

As can be seen from the graph, superior attenuation is provided by thecoders in accordance with the invention, in the ranges where f/f_(s)≲0.2, i.e., where there is oversampling. This is true notwithstandingthe use, in prior art apparatus, of intentionally "leaky" integrators,which tend to reduce but not eliminate entirely the peak at the transferfunction pole.

The invention described above with respect to predictive coders is alsoapplicable to error feedback coders, such as prior art coders of thetype depicted in FIG. 7. As shown, the forward loop includes an A/Dconverter or quantizer 701, and the feedback loop contains a feedbackfilter 702. However, in this arrangement, the input to quantizer 701,which is a composite signal x_(n) comprising the difference between theinput signal s_(n) and a feedback signal f_(n), is subtracted from theoutput thereof by a differencing circuit 704, to form a quantizing errorsignal q_(n). The error signal q_(n) is filtered by filter 702 to formthe aforesaid feedback signal, which is then subtracted from the inputsignal in a second differencing circuit 705. In this arrangement,assuming that filter 702 is an N tap delay line with coefficients a₁ . .. a_(n), the quantizer 701 output signal X_(n) is the nearest discretequantization level to the quantizer input sample x_(n), wherein##EQU14## and

    X.sub.n = x.sub.n + q.sub.n .                              (21)

Since, as stated previously, equation (6) is the transfer function offilter 702, equation (20) may be written in Z - transform rotation as

    X(Z) = S(Z) + [1-H(Z)]·q(Z) .                     (22)

equation (22) shows that the spectral density of the quantizing noise isshaped by the response of the Z-transform 1-H(Z), and thus thatappropriate choice of the coefficients a_(i), assuming f_(s) >>f_(o),will minimize the amount of in-band noise power by placing most of thenoise out-of-band where it may be eliminated using a digital low passfilter. By so doing, the quantization level separation can be increasedrelative to that of an A/D converter without feedback; this, in turn,permits a desired decrease in the number of quantization levelsrequired.

Minimization of noise in the band between -f_(o) and f_(o) is generallyaccomplished, using prior art techniques, by first recognizing that thenoise power Q(f_(o)) is given by ##EQU15## where G_(q) (f) is thetwo-sided power density function of the individual quantizing errorsq_(n), which, it is assumed, are not correlated with the input samples,and where ##EQU16## Expanding equation (23) using equation (24) gives##EQU17## where ##EQU18## Filter coefficients which minimize Q(f_(o))may then be obtained by solving the simultaneous equations

    ∂Q(f.sub.o)/∂a.sub.k = 0 K = 1,2 . . . N, (27)

which yields a solution

    B =  Γ.sup.-1 . γ                              (28)

where B and γ are vectors with components a_(k) and β_(k), respectively,and Γ is an NXM matrix with components

    Γ.sub.j,k = β.sub.j-k .                         (29)

This solution, however, is often undesirable in that the coefficientsobtained from equation (29) depend on the noise spectral density G_(q)(f).

In accordance with the invention, spectrum independent coefficients areobtained in a manner similar to that described above for predictivecoders, namely by expanding equation (25) as a power series in (f_(o)/f_(s))² and choosing the a_(k) such that coefficients of terms up tothe N^(th) term vanish. The resulting coefficients are again given by##EQU19## and the transfer function of an N-tap delay line withcoefficients according to equation (30) is again given by equation (14).

Realization of an error feedback pulse code modulation encoder utilizinganalog integrators in lieu of an N tap delay line in the feedback loopis accomplished as shown in FIG. 8. As in FIG. 7, the forward loopincludes a quantizer 801 which is arranged to sample the compositesignal X(Z) applied thereto at a rate f_(s) greater than the Nyquistrate 2f_(o). However, the input signal is applied to quantizer 801 via afirst N^(th) order integrator 806, followed by a combinatorial circuit805. Circuit 805 is arranged to algebraically combine (i.e., add orsubtract in a desired manner) the input signals applied thereto. In thefeedback loop, the output of quantizer 801 is applied to a digital toanalog converter 802, which may be constructed with the same simplicityas quantizer 801, to form a direct feedback signal f_(n). This signal isapplied via line 804 to a positive input of circuit 805, and also to theinput of a second N^(th) order integrator 803, the output of which isapplied to a negative input of circuit 805. N^(th) order integrators 803and 806 are simply serial chains of N analog integrators. For example, athird order integrator would simply include integrators 901, 902 and903, as shown in FIG. 9.

The equivalence between the coders of FIGS. 7 and 8 when the formeradvantageously utilizes the coefficients of equation (30) may beappreciated by rewriting equation (22) as

    X(Z) = S(Z) + G(Z) ·q(Z)                          (31)

wherein

    G(Z) .tbd. 1-H(Z) .                                        (32)

bearing in mind that the transfer function of a multistage delay linewith coefficients in accordance with equation (30) is

    H(Z) = 1-(1-Z.sup.-1).sup.N,                               (14)

then

    G(Z) = (1-Z.sup.-1).sup.N .

accordingly, the transfer function of N^(th) order integrators 803 amnd806 is given by 1/G(Z), since, as stated previously, the transferfunction of an individual integrator is 1/(1-Z⁻¹). Applying the foregingto the coder FIG. 8, it will be seen that for an input signal S(Z), andan output signal X(Z), the output of integrator 806 is given byS(Z)/G(Z), the output of integrator 803 is given by X(Z)/G(Z), and thesignal on line 804 is the output signal X(Z). The output of circuit 805is thus ##EQU20## and the output of quantizer 801 is thus ##EQU21##Equation (34) when simpified, results in equation (31), as desired.

The advantages of the coder of FIG. 8 can be appreciated by reference toFIG. 10, which shows the reduction in the number of quantization levelsachievable by use of the present invention. For a sampling rate f_(s) of20 times the highest signal frequency f_(o), and for a three stageintegrator, a reduction in the number of levels by a factor of 2⁸ isattained.

While the above descriptions of predictive and error feedback codersindicate that performance improves as the number N of delay line stages(or integrators) increases, this theoretical improvement is limited byexperimental implementations which indicate that use of feedbacknetworks in which N>5 is not practical. First, the loop gain G isnominally unity but is bounded, for stability reasons by

    2.sup.N /(2.sup.N +1) < G < 2.sup.N /(2.sup.N -1)          (35)

for an N-tap predictor. Thus, as N gets larger, the stability limits arenarrowed. Furthermore, loop delay is added by the nonzero propagationdelays of the A/D and D/A converters and must be compensated by, forexample, feedforward techniques in the cascaded integrators of FIG. 5.Neglecting such compensation can also lead to coder instability.Therefore, the simplification of the quantizer is offset by anincreasing sensitivity of the coder stability to the analog feedbacknetwork as N increases.

In addition to increased stability problems as N becomes large, thequantizer input signal power contains a contribution, neglected above,from the nonzero quantizing errors. Letting σ² _(e) be the mean squarepower of the predictive coder's difference signal e_(n), σ² _(s) be themean square power of the error feedback coder's input signal power andassuming a uniform level separation δ in the quantizer (randomquantizing errors and noise power δ² /12), the mean square value σ_(in)² of the quantizer input signal is ##EQU22## for predictive coders and##EQU23## for error feedback coders. In predictive coders, σ² _(e)decreases with N increasing while δ₂ remains fixed, setting a lowerlimit on δ_(in) ² as N increases. With error feedback coders, δincreases with increasing N while σ_(s) ² remains fixed, again placing alower limit on σ_(in) ².

For these reasons, a practical limit on feedback filter complexity in anoversampled coder realization is likely to be N ≲ 5.

An example of the advantages of the present invention is illustrated bythe A/D interface unit which is presently employed to quantize thefrequency division multiplexed signals in the digital processordescribed in Communication Technology, Vol. COM-19, No. 6, December 1971pp. 1050-59. This unit presently quantizes a 108 KHz bandwidth signalusing a 13 bit uniform PCM code. However, utilizing a 4 stage predictivecoder constructed in accordance with the invention, with a 2 MHz samplerate, only an 8-level (3 bit) quantizer is required.

While not shown in the drawing, remote decoding of the outputs of boththe predictive and error feedback coders of the present invention isquite simply attained. For predictive encoders, the output/inputrelationship for the entire coder is given by

    O(Z) = (1-Z.sup.-1).sup.N [I(Z) + q(Z)],                   (38)

where O(Z) is the output signal, I(Z) is the input signal and q(Z) isthe quantizing error. Accordingly, the remote decoder may comprise aserial chain of N integrators, which, as stated previously, has atransfer function given by 1/(1-Z⁻¹)^(N). In the case of error feedbackencoders, the output/input relationship is given by

    O(Z) = I(Z) + (I-Z.sup.- 1).sup.N q(Z) .                   (39)

accordingly, a low pass filter with a stop band chosen to compensate forthe increasing noise spectral density at higher frequencies, mayconveniently be used for decoding.

Various modifications and adaptations of the present invention will bereadily apparent to those skilled in the art. For this reason, it isintended the invention be limited only by the appended claims.

What is claimed is:
 1. An error feedback pulse code modulation encoderfor converting an analog input signal to a digital output signalrepresentative thereof, comprisinga first differencing circuit forcombining said input signal and an error feedback signal to generate acomposite signal, quantizing means responsive to said first differencingcircuit for generating said digital output signal by sampling saidcomposite signal and converting said samples to digital form, a seconddifferencing circuit for combining said composite signal and saiddigital output signal to generate a quantizing error signal, and meansfor generating said error feedback signal from said quantizing errorsignal including
 1. a delay line having stages 1 through n for receivingsaid quantizing error signal, each of said stages having an output tap,2. n attenuators having attenuation coefficients a₁ through a_(n)connected to respective ones of said output taps, and
 3. means foralgebraically summing the outputs of said attenuators to form said errorfeedback signal,wherein said attenuation coefficients are selected inaccordance with the relationship ##EQU24##
 2. The invention defined inclaim 1 wherein said analog signal has a bandwidth f_(o) and saidquantizer is arranged to sample said error signal at a rate of at least5 f_(o).
 3. An error feedback pulse code modulation encoder arranged toconvert an analog input signal to a digital output signal representativethereof, comprisinga quantizer for sampling a composite analog signalapplied thereto at a rate f_(s) to generate said digital output signal,means for generating a quantization error signal from the differencebetween digital output signal and said composite signal, a filter forreceiving said quantization error signal and for generating a feedbacksignal therefrom, and means for generating said composite signal fromthe difference between said analog input signal and said feedbacksignal, wherein said filter has a transfer function K(Z) in the Z domaingiven by

    K(Z) = 1-(1-Z.sup.-1).sup.N

and N is an integer between 2 and 5.